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Theorem fvreseq 5630
Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fvreseq  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Distinct variable groups:    x, B    x, F    x, G
Allowed substitution hint:    A( x)

Proof of Theorem fvreseq
StepHypRef Expression
1 fnssres 5359 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fnssres 5359 . . . 4  |-  ( ( G  Fn  A  /\  B  C_  A )  -> 
( G  |`  B )  Fn  B )
31, 2anim12i 549 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  ( G  Fn  A  /\  B  C_  A
) )  ->  (
( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B ) )
43anandirs 804 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B
) )
5 eqfnfv 5624 . . 3  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( ( F  |`  B ) `  x )  =  ( ( G  |`  B ) `
 x ) ) )
6 fvres 5544 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
7 fvres 5544 . . . . 5  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
86, 7eqeq12d 2299 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  =  ( ( G  |`  B ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
98ralbiia 2577 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
)  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
105, 9syl6bb 252 . 2  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
114, 10syl 15 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545    C_ wss 3154    |` cres 4693    Fn wfn 5252   ` cfv 5257
This theorem is referenced by:  tfrlem1  6393  tfr3  6417  fseqenlem1  7653  dchrresb  20500  rdgprc  24153  predreseq  24181  wfr3g  24257  frr3g  24282  bnj1536  28959  bnj1253  29120  bnj1280  29123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265
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